Camassa-Holm Equations as a Closure Model for Turbulent Channel and Pipe Flow

Chen, Shiyi; Foiaş, Ciprian; Holm, Darryl D.; Olson, Eric J.; Titi, Edriss S.; Wynne, Shannon

doi:10.1103/physrevlett.81.5338

Cited by 310 publications

(335 citation statements)

References 19 publications

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“…However, the approach used historically in Chen et al [11] and Holm and Titi [12] for deriving the closed Eulerian form (1.3) of the LANS-α motion equation was based on the combination of two other earlier results. First, the Lagrangian-averaged variational principle of Gjaja and Holm [13] was applied for deriving the inviscid averaged nonlinear fluid equations, obtained by averaging Hamilton's principle for fluids over the rapid phase of their small turbulent circulations at fixed Lagrangian coordinate: this step had its own precedent in earlier work on Lagrangianaveraged fluid equations by Andrews and McIntyre [14].…”

Section: As Expected the Lans-α Motion Equation Satisfies Thementioning

confidence: 99%

Length-scale estimates for the LANS- equations in terms of the Reynolds number

Gibbon

Holm

2006

Physica D: Nonlinear Phenomena

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(2002) 289-306] for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number Re, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. Re is defined as Re = U /ν where U is a bounded spatio-temporally averaged Navier-Stokes velocity field and the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by λ −1 k ≤ c ( /α) 1/4 Re 5/8 . Moreover, the estimate of Foias, Holm and Titi for the fractal dimension of the global attractor, in terms of Re, comes out to bewhere V α = L/( α) 1/2 3 and V = (L/ ) 3 . It is also shown that there exists a series of time-averaged inverse squared length scales whose members, κ 2 n,0 , are estimated as (n ≥ 1)The upper bound on the first member of the hierarchy κ 2 1,0 coincides with the inverse squared Taylor micro-scale to within log-corrections.

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Section: As Expected the Lans-α Motion Equation Satisfies Thementioning

confidence: 99%

Length-scale estimates for the LANS- equations in terms of the Reynolds number

Gibbon

Holm

2006

Physica D: Nonlinear Phenomena

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“…The general idea is to regularize the fluid equations (Navier-Stokes or MHD) at the level of the Euler-Poincaré variational principle, by adding terms into the Lagrangian that include gradients of the fluid velocity. These terms penalize the formation of small scale structures, and can thus be used as a large eddy simulation (LES) model, causing turbulence to dissipate at larger scales 56 . These methods have been shown to have some significant advantages over more traditional LES methods (for instance those based on hyperdiffusion) especially for simulation of MHD turbulence 37,57 .…”

Section: Numerical Applicationsmentioning

confidence: 99%

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Squire¹,

Qin²,

Tang³

2012

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We present a new variational principle for the gyrokinetic system, similar to the MaxwellVlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincaré theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods.

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“…The first model we consider is the Lagrangian-averaged Navier-Stokes (LANS) α-model [10,25]. It is derived by Lagrangian averaging fluid motions followed by application of Taylor's frozen-in turbulence approximation as the model's one and only closure: fluctuations about the Lagrangian mean smaller than α are swept along by the large-scale flow and are not allowed to interact with one another [24].…”

Section: Lans-α and Rigid Body Formationmentioning

confidence: 99%

“…Regularization modeling (of the SFS stress tensor) for Navier-Stokes [8,10,14,17,18,25,29,36,50], magnetohydrodynamics (MHD) [24], Boussinesq convection [53], and inviscid cases [33] promises several advantages. For Navier-Stokes, only weak, possibly non-unique solutions have been rigorously proven to exist, and this can impact the possibility of achieving a direct numerical solution (DNS), e.g., with Fourier methods [22].…”

Section: Introductionmentioning

confidence: 99%

The Effect of Subfilter-Scale Physics on Regularization Models

et al. 2010

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The subfilter-scale (SFS) physics of regularization models are investigated to understand the regularizations' performance as SFS models. Suppression of spectrally local SFS interactions and conservation of small-scale circulation in the Lagrangian-averaged Navier-Stokes α-model (LANS-α) is found to lead to the formation of rigid bodies. These contaminate the superfilter-scale energy spectrum with a scaling that approaches k +1 as the SFS spectra is resolved. The Clark-α and Leray-α models, truncations of LANS-α, do not conserve small-scale circulation and do not develop rigid bodies. LANS-α, however, is closest to Navier-Stokes in intermittency properties. All three models are found to be stable at high Reynolds number. Differences between L 2 and H 1 norm models are clarified. For magnetohydrodynamics (MHD), the presence of the Lorentz force as a source (or sink) for circulation and as a facilitator of both spectrally nonlocal large to small scale interactions as well as local SFS interactions prevents the formation of rigid bodies in Lagrangianaveraged MHD (LAMHD-α). LAMHD-α performs well as a predictor of superfilter-scale energy spectra and of intermittent current sheets at high Reynolds numbers. It may prove generally applicable as a MHD-LES.

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Camassa-Holm Equations as a Closure Model for Turbulent Channel and Pipe Flow

Cited by 310 publications

References 19 publications

Length-scale estimates for the LANS- equations in terms of the Reynolds number

Length-scale estimates for the LANS- equations in terms of the Reynolds number

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

The Effect of Subfilter-Scale Physics on Regularization Models

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